A dcpo-completion of a poset \(P\)  is a dcpo \(A\)  together with a Scott-continuous mapping \(\eta :P \longrightarrow A\)  such that for any Scott-continuous mapping \(f:P \longrightarrow B\)  into a dcpo \(B\)  there exists a unique Scott-continuous mapping \(\hat{f}:A \longrightarrow B\)  satisfying \(f = \hat{f} \circ \eta \) . It was shown in [1]} that the dcpo-completion, \(E(P)\) , of a poset \(P\)  always exists; furthermore \(\Gamma (P) \cong \Gamma (E(P))\) . Hence the class, \(\mathbf {POS}\) , of all posets is not \(\Gamma \) -faithful. Indeed any class of posets that is strictly larger than \(\mathbf {DCPO}\) , the class of dcpos, is not \(\Gamma \) -faithful. This means that we can restrict attention entirely to dcpos, in particular, it leads one to ask:
([2]}, [1]})

Is \(\mathbf {DCPO}\)  \(\Gamma \) -faithful?

This question was dubbed the Ho-Zhao problem in [4]}. The authors of this paper claimed that the two dcpos
\(\Upsilon = ([0,1],\le ) \text{ and } \Psi = (\lbrace [0,a] \mid 0 < a \le 1\rbrace ,\subseteq )\) 
